Integrand size = 20, antiderivative size = 139 \[ \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx=-\frac {a^5 d}{x}+a^5 e x+\frac {5}{3} a^4 c d x^3+a^4 c e x^5+\frac {10}{7} a^3 c^2 d x^7+\frac {10}{9} a^3 c^2 e x^9+\frac {10}{11} a^2 c^3 d x^{11}+\frac {10}{13} a^2 c^3 e x^{13}+\frac {1}{3} a c^4 d x^{15}+\frac {5}{17} a c^4 e x^{17}+\frac {1}{19} c^5 d x^{19}+\frac {1}{21} c^5 e x^{21} \]
-a^5*d/x+a^5*e*x+5/3*a^4*c*d*x^3+a^4*c*e*x^5+10/7*a^3*c^2*d*x^7+10/9*a^3*c ^2*e*x^9+10/11*a^2*c^3*d*x^11+10/13*a^2*c^3*e*x^13+1/3*a*c^4*d*x^15+5/17*a *c^4*e*x^17+1/19*c^5*d*x^19+1/21*c^5*e*x^21
Time = 0.01 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx=-\frac {a^5 d}{x}+a^5 e x+\frac {5}{3} a^4 c d x^3+a^4 c e x^5+\frac {10}{7} a^3 c^2 d x^7+\frac {10}{9} a^3 c^2 e x^9+\frac {10}{11} a^2 c^3 d x^{11}+\frac {10}{13} a^2 c^3 e x^{13}+\frac {1}{3} a c^4 d x^{15}+\frac {5}{17} a c^4 e x^{17}+\frac {1}{19} c^5 d x^{19}+\frac {1}{21} c^5 e x^{21} \]
-((a^5*d)/x) + a^5*e*x + (5*a^4*c*d*x^3)/3 + a^4*c*e*x^5 + (10*a^3*c^2*d*x ^7)/7 + (10*a^3*c^2*e*x^9)/9 + (10*a^2*c^3*d*x^11)/11 + (10*a^2*c^3*e*x^13 )/13 + (a*c^4*d*x^15)/3 + (5*a*c^4*e*x^17)/17 + (c^5*d*x^19)/19 + (c^5*e*x ^21)/21
Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1585, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+c x^4\right )^5 \left (d+e x^2\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 1585 |
\(\displaystyle \int \left (\frac {a^5 d}{x^2}+a^5 e+5 a^4 c d x^2+5 a^4 c e x^4+10 a^3 c^2 d x^6+10 a^3 c^2 e x^8+10 a^2 c^3 d x^{10}+10 a^2 c^3 e x^{12}+5 a c^4 d x^{14}+5 a c^4 e x^{16}+c^5 d x^{18}+c^5 e x^{20}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 d}{x}+a^5 e x+\frac {5}{3} a^4 c d x^3+a^4 c e x^5+\frac {10}{7} a^3 c^2 d x^7+\frac {10}{9} a^3 c^2 e x^9+\frac {10}{11} a^2 c^3 d x^{11}+\frac {10}{13} a^2 c^3 e x^{13}+\frac {1}{3} a c^4 d x^{15}+\frac {5}{17} a c^4 e x^{17}+\frac {1}{19} c^5 d x^{19}+\frac {1}{21} c^5 e x^{21}\) |
-((a^5*d)/x) + a^5*e*x + (5*a^4*c*d*x^3)/3 + a^4*c*e*x^5 + (10*a^3*c^2*d*x ^7)/7 + (10*a^3*c^2*e*x^9)/9 + (10*a^2*c^3*d*x^11)/11 + (10*a^2*c^3*e*x^13 )/13 + (a*c^4*d*x^15)/3 + (5*a*c^4*e*x^17)/17 + (c^5*d*x^19)/19 + (c^5*e*x ^21)/21
3.1.6.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && IGtQ[q, -2]
Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {a^{5} d}{x}+a^{5} e x +\frac {5 a^{4} c d \,x^{3}}{3}+a^{4} c e \,x^{5}+\frac {10 a^{3} c^{2} d \,x^{7}}{7}+\frac {10 a^{3} c^{2} e \,x^{9}}{9}+\frac {10 a^{2} c^{3} d \,x^{11}}{11}+\frac {10 a^{2} c^{3} e \,x^{13}}{13}+\frac {a \,c^{4} d \,x^{15}}{3}+\frac {5 a \,c^{4} e \,x^{17}}{17}+\frac {c^{5} d \,x^{19}}{19}+\frac {c^{5} e \,x^{21}}{21}\) | \(122\) |
risch | \(-\frac {a^{5} d}{x}+a^{5} e x +\frac {5 a^{4} c d \,x^{3}}{3}+a^{4} c e \,x^{5}+\frac {10 a^{3} c^{2} d \,x^{7}}{7}+\frac {10 a^{3} c^{2} e \,x^{9}}{9}+\frac {10 a^{2} c^{3} d \,x^{11}}{11}+\frac {10 a^{2} c^{3} e \,x^{13}}{13}+\frac {a \,c^{4} d \,x^{15}}{3}+\frac {5 a \,c^{4} e \,x^{17}}{17}+\frac {c^{5} d \,x^{19}}{19}+\frac {c^{5} e \,x^{21}}{21}\) | \(122\) |
norman | \(\frac {-d \,a^{5}+a^{5} e \,x^{2}+\frac {5}{3} a^{4} c d \,x^{4}+a^{4} c e \,x^{6}+\frac {10}{7} a^{3} c^{2} d \,x^{8}+\frac {10}{9} a^{3} c^{2} e \,x^{10}+\frac {10}{11} a^{2} c^{3} d \,x^{12}+\frac {10}{13} a^{2} c^{3} e \,x^{14}+\frac {1}{3} a \,c^{4} d \,x^{16}+\frac {5}{17} a \,c^{4} e \,x^{18}+\frac {1}{19} c^{5} d \,x^{20}+\frac {1}{21} c^{5} e \,x^{22}}{x}\) | \(125\) |
gosper | \(-\frac {-138567 c^{5} e \,x^{22}-153153 c^{5} d \,x^{20}-855855 a \,c^{4} e \,x^{18}-969969 a \,c^{4} d \,x^{16}-2238390 a^{2} c^{3} e \,x^{14}-2645370 a^{2} c^{3} d \,x^{12}-3233230 a^{3} c^{2} e \,x^{10}-4157010 a^{3} c^{2} d \,x^{8}-2909907 a^{4} c e \,x^{6}-4849845 a^{4} c d \,x^{4}-2909907 a^{5} e \,x^{2}+2909907 d \,a^{5}}{2909907 x}\) | \(128\) |
parallelrisch | \(\frac {138567 c^{5} e \,x^{22}+153153 c^{5} d \,x^{20}+855855 a \,c^{4} e \,x^{18}+969969 a \,c^{4} d \,x^{16}+2238390 a^{2} c^{3} e \,x^{14}+2645370 a^{2} c^{3} d \,x^{12}+3233230 a^{3} c^{2} e \,x^{10}+4157010 a^{3} c^{2} d \,x^{8}+2909907 a^{4} c e \,x^{6}+4849845 a^{4} c d \,x^{4}+2909907 a^{5} e \,x^{2}-2909907 d \,a^{5}}{2909907 x}\) | \(128\) |
-a^5*d/x+a^5*e*x+5/3*a^4*c*d*x^3+a^4*c*e*x^5+10/7*a^3*c^2*d*x^7+10/9*a^3*c ^2*e*x^9+10/11*a^2*c^3*d*x^11+10/13*a^2*c^3*e*x^13+1/3*a*c^4*d*x^15+5/17*a *c^4*e*x^17+1/19*c^5*d*x^19+1/21*c^5*e*x^21
Time = 0.25 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx=\frac {138567 \, c^{5} e x^{22} + 153153 \, c^{5} d x^{20} + 855855 \, a c^{4} e x^{18} + 969969 \, a c^{4} d x^{16} + 2238390 \, a^{2} c^{3} e x^{14} + 2645370 \, a^{2} c^{3} d x^{12} + 3233230 \, a^{3} c^{2} e x^{10} + 4157010 \, a^{3} c^{2} d x^{8} + 2909907 \, a^{4} c e x^{6} + 4849845 \, a^{4} c d x^{4} + 2909907 \, a^{5} e x^{2} - 2909907 \, a^{5} d}{2909907 \, x} \]
1/2909907*(138567*c^5*e*x^22 + 153153*c^5*d*x^20 + 855855*a*c^4*e*x^18 + 9 69969*a*c^4*d*x^16 + 2238390*a^2*c^3*e*x^14 + 2645370*a^2*c^3*d*x^12 + 323 3230*a^3*c^2*e*x^10 + 4157010*a^3*c^2*d*x^8 + 2909907*a^4*c*e*x^6 + 484984 5*a^4*c*d*x^4 + 2909907*a^5*e*x^2 - 2909907*a^5*d)/x
Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx=- \frac {a^{5} d}{x} + a^{5} e x + \frac {5 a^{4} c d x^{3}}{3} + a^{4} c e x^{5} + \frac {10 a^{3} c^{2} d x^{7}}{7} + \frac {10 a^{3} c^{2} e x^{9}}{9} + \frac {10 a^{2} c^{3} d x^{11}}{11} + \frac {10 a^{2} c^{3} e x^{13}}{13} + \frac {a c^{4} d x^{15}}{3} + \frac {5 a c^{4} e x^{17}}{17} + \frac {c^{5} d x^{19}}{19} + \frac {c^{5} e x^{21}}{21} \]
-a**5*d/x + a**5*e*x + 5*a**4*c*d*x**3/3 + a**4*c*e*x**5 + 10*a**3*c**2*d* x**7/7 + 10*a**3*c**2*e*x**9/9 + 10*a**2*c**3*d*x**11/11 + 10*a**2*c**3*e* x**13/13 + a*c**4*d*x**15/3 + 5*a*c**4*e*x**17/17 + c**5*d*x**19/19 + c**5 *e*x**21/21
Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx=\frac {1}{21} \, c^{5} e x^{21} + \frac {1}{19} \, c^{5} d x^{19} + \frac {5}{17} \, a c^{4} e x^{17} + \frac {1}{3} \, a c^{4} d x^{15} + \frac {10}{13} \, a^{2} c^{3} e x^{13} + \frac {10}{11} \, a^{2} c^{3} d x^{11} + \frac {10}{9} \, a^{3} c^{2} e x^{9} + \frac {10}{7} \, a^{3} c^{2} d x^{7} + a^{4} c e x^{5} + \frac {5}{3} \, a^{4} c d x^{3} + a^{5} e x - \frac {a^{5} d}{x} \]
1/21*c^5*e*x^21 + 1/19*c^5*d*x^19 + 5/17*a*c^4*e*x^17 + 1/3*a*c^4*d*x^15 + 10/13*a^2*c^3*e*x^13 + 10/11*a^2*c^3*d*x^11 + 10/9*a^3*c^2*e*x^9 + 10/7*a ^3*c^2*d*x^7 + a^4*c*e*x^5 + 5/3*a^4*c*d*x^3 + a^5*e*x - a^5*d/x
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx=\frac {1}{21} \, c^{5} e x^{21} + \frac {1}{19} \, c^{5} d x^{19} + \frac {5}{17} \, a c^{4} e x^{17} + \frac {1}{3} \, a c^{4} d x^{15} + \frac {10}{13} \, a^{2} c^{3} e x^{13} + \frac {10}{11} \, a^{2} c^{3} d x^{11} + \frac {10}{9} \, a^{3} c^{2} e x^{9} + \frac {10}{7} \, a^{3} c^{2} d x^{7} + a^{4} c e x^{5} + \frac {5}{3} \, a^{4} c d x^{3} + a^{5} e x - \frac {a^{5} d}{x} \]
1/21*c^5*e*x^21 + 1/19*c^5*d*x^19 + 5/17*a*c^4*e*x^17 + 1/3*a*c^4*d*x^15 + 10/13*a^2*c^3*e*x^13 + 10/11*a^2*c^3*d*x^11 + 10/9*a^3*c^2*e*x^9 + 10/7*a ^3*c^2*d*x^7 + a^4*c*e*x^5 + 5/3*a^4*c*d*x^3 + a^5*e*x - a^5*d/x
Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx=\frac {c^5\,d\,x^{19}}{19}-\frac {a^5\,d}{x}+\frac {c^5\,e\,x^{21}}{21}+a^5\,e\,x+\frac {10\,a^3\,c^2\,d\,x^7}{7}+\frac {10\,a^2\,c^3\,d\,x^{11}}{11}+\frac {10\,a^3\,c^2\,e\,x^9}{9}+\frac {10\,a^2\,c^3\,e\,x^{13}}{13}+\frac {5\,a^4\,c\,d\,x^3}{3}+\frac {a\,c^4\,d\,x^{15}}{3}+a^4\,c\,e\,x^5+\frac {5\,a\,c^4\,e\,x^{17}}{17} \]